Early Exercise





Kerry Back

Intrinsic value

  • The “intrinsic value” of an option is what it would be worth if it were immediately expiring.
    • The intrinsic value of a call is max(0, stock price - strike)
    • The intrinsic value of a put is max(0, strike - stock price)
  • Prior to maturity, the value of an option usually exceeds its intrinsic value. It is “better alive than dead.”
  • No one should exercise when value > intrinsic value

Early exercise

  • Sometimes, prior to maturity, the value of an option equals its the intrinsic value.
  • All options should be exercised, or positions should be closed by trading.

Calls are better alive than dead

  • A call option on an underlying asset that does not pay dividends prior to the option maturity should never be exercised early.
  • Proof is by put-call parity:

\[ \text{call price} = \text{stock price} + \text{put price} - \text{PV of strike}\]

\[ \text{call price} > \text{stock price} - \text{PV of strike}\]

\[ \text{call price} > \text{stock price} - \text{strike}\]

Possible counter-argument

  • You own a call option and think the stock will fall.
  • Should you exercise the call and sell the stock before it falls instead of holding on to the option?
  • No, instead: keep the option alive and short the stock.
    • Stock goes up \(\Rightarrow\) happy you kept the call
    • Stock goes down \(\Rightarrow\) cover short with call
  • This assumes you (or someone in the market) can short. It does not apply to executive stock options, which often should be exercised early.

Early exercise of puts is often optimal

  • If the stock price has fallen very low, you can be pretty sure you will not regret exercising a put.
  • Exercising the put early gets the cash early, which can be invested.
  • There is always a critical stock price (threshold) such that it is optimal to exercise a put when the stock falls below the threshold.
  • The threshold depends crucially on the stock volatility.